Root test fails, now what?

3. The attempt at a solution
So I did the root test first, but the limit on the inside is 1. I then tried the ratio test but then when I tried taking the limit, I ended up with e^(-infinity+infinity). Finally, I tried the test for divergence on the original series, but i turned out be zero so it might or might not converge. Now I'm totally stuck.

Staff: Mentor

3. The attempt at a solution
So I did the root test first, but the limit on the inside is 1. I then tried the ratio test but then when I tried taking the limit, I ended up with e^(-infinity+infinity).

You should never end up with something like this ( -∞ + ∞), as it is indeterminate. I agree with what you got on the root test, so that's not a useful test, and the n-th term test seems to give a value of 0, so it doesn't tell us anything.

I would take another look at the ratio test to see what the limit actually is.

3. The attempt at a solution
So I did the root test first, but the limit on the inside is 1. I then tried the ratio test but then when I tried taking the limit, I ended up with e^(-infinity+infinity). Finally, I tried the test for divergence on the original series, but i turned out be zero so it might or might not converge. Now I'm totally stuck.

If
[tex]t_n = \left(1 - \frac{1}{n^{1/3}}\right)^n,[/tex]
you can look at ##L_n = \ln(t_n)## and use the series expansion of ##\ln (1-x)## for small ##x = 1/n^{1/3}##, to get the behavior of ##t_n## for large n. In fact, you can even get a simple upper bound ##u_n##, so that ## 0 < t_n < u_n##, and ##\sum u_n## is easy to analyze.